# A Survey of Lattice-Based Physical-Layer Security for Wireless Systems with p-Modular Lattice Constructions

## TL;DR

This paper surveys lattice-based methods for securing wireless communications, focusing on algebraic constructions and new p-modular lattices.

## Contribution

The paper introduces a new family of p-modular lattices and analyzes their properties for physical-layer security.

## Key findings

- p-modular lattices from cyclotomic fields Q(ζp) for primes p≡1(mod4) are characterized and analyzed.
- A non-existence theorem shows these lattices cannot be extended to prime-power cyclotomic fields.
- Indefinite theta series and modular completions are integrated to support secrecy metrics in mixed-signature lattices.

## Abstract

Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and fading wiretap channels. This article offers a comprehensive survey of lattice-based wiretap coding, covering foundational concepts in algebraic number theory, Construction A over number fields, and the structure of modular and unimodular lattice families. We review key secrecy metrics, including secrecy gain, flatness factor, and equivocation, and consolidate classical and recent results to provide a unified perspective that links wireless-channel models with their underlying algebraic lattice structures. In addition, we review a newly proposed family of p-modular lattices in Khodaiemehr, H., 2018 constructed from cyclotomic fields Q(ζp) for primes p≡1(mod4) via a generalized Construction A framework. We characterize their algebraic and geometric properties and establish a non-existence theorem showing that such constructions cannot be extended to prime-power cyclotomic fields Q(ζpn) with n>1. Finally, motivated by the fact that these p-modular lattices naturally yield mixed-signature structures for which classical theta series diverge, we integrate recent advances on indefinite theta series and modular completions. Drawing on Vignéras’ differential framework and generalized error functions, we outline how modularly completed indefinite theta series provide a principled analytic foundation for defining secrecy-relevant quantities in the indefinite setting. Overall, this work serves both as a survey of algebraic lattice techniques for PLS and as a source of new design insights for secure wireless communication systems.